Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces |
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Authors: | Xiaolong Qin Yongfu Su Meijuan Shang |
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Institution: | (1) Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, PR China;(2) Department of Mathematics, Shijiazhuang University, Shijiazhuang, 050035, PR China |
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Abstract: | Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T
1, T
2 and T
3: K → E be asymptotically nonexpansive mappings with {k
n
}, {l
n
} and {j
n
}. 1, ∞) such that Σ
n=1
∞
(k
n
− 1) < ∞, Σ
n=1
∞
(l
n
− 1) < ∞ and Σ
n=1
∞
(j
n
− 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T
1x
= T
2x
= T
3
x} = x} denotes the common fixed points set of T
1, T
2 and T
3. Let {α
n
}, {α′
n
} and {α″
n
} be real sequences in (0, 1) and ∈ ≤ {α
n
}, {α′
n
}, {α″
n
} ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x
1 ∈ K define the sequence {x
n
} by (i) If the dual E* of E has the Kadec-Klee property then {x
n
} converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then {x
n
} converges strongly to a common fixed point p ∈ F.
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Keywords: | Asymptotically nonexpansive non-self map Kadec-Klee property Uniformly convex |
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